Firstly, your query string is incorrect. It should be:
https://www.hoas.fi/web/hak_inet.nsf/WebHakemus?OpenForm&02.07&PersonFirstName=Alex&PersonLastName=Smith
Secondly, in order to pre-populate the page with the results of the query string, the developer of the page would have had to added logic to extract the query string values and pre-populate the page with those values. In this case, it does not appear that they have done that.
You could try saving the page locally as HTML. Then you could modify the HTML to include your default values. You would also need to update any relative paths to point to the server as a full URL. Then you could open the page on your machine and hopefully post to the server. This assumes that they are not injecting any session or other temporary information in the page that they validate.